Drawing A2xA2 Dynkin diagram with dynkin-diagrams

Question

This is a follow up question: How do I draw it exactly as in the blackboard picture I provide, that is with ellipses around the nodes and for the bottom one with the connections to the rightmost node straight not curved.

I posted it as a comment first but I am not sure people notice when comments are added. Thanks a lot.

Edit: here is minimal working example

\documentclass[12pt]{amsbook}

\usepackage{dynkin-diagrams}

\begin{document}
 
 Question: how do I produced a diagram which only keeps the square in this diagram 
 \dynkin[fold] A{*****}? I.e. a Satake-Tits diagram of type A2xA2 but I do not want to have just \dynkin A2 \dynkin A2 but have the nodes paired up. 

\end{document}

---

I am using the excellent package Dynkin-diagram .sty and drawing the diagrams for the non-split version of the Freudenthal-Tits magic square. I managed to draw all but one of them. In picture below the bottom one can easily be drawn with

\dynkin[fold] A{*****}

My question is how do I draw the top one? Thanks a lot

Addition: Benjamin Kindly offered to include FTMS non-split into his package. The others I managed to draw, here the output as well as latex code which produces it

\documentclass[12pt]{amsbook}
\usepackage[dvipsnames,table]{xcolor}
\usepackage{dynkin-diagrams}


\begin{document}
 
\begin{center}
\Large
\begin{tabular}{| c | c | c | c | c |}\hline
\cellcolor{lightgray!90} $\mathbb{A}\setminus \mathbb{B}$ &  \cellcolor{lightgray!90} $\mathbb{K}$  & \cellcolor{lightgray!90} $\mathbb{L}$   & \cellcolor{lightgray!90} $\mathbb{H}$ & \cellcolor{lightgray!90} $\mathbb{O}$  \\ \hline
\cellcolor{lightgray!90} $\mathbb{K}$ &  \cellcolor{BurntOrange!80} \dynkin A1  & \cellcolor{BurntOrange!80}  \dynkin A{*o}   & \cellcolor{BurntOrange!80} \dynkin C{o*o} & \cellcolor{BurntOrange!80} \dynkin F{*ooo}  \\ \hline
\cellcolor{lightgray!90} $\mathbb{L}$ &  \cellcolor{SeaGreen} \dynkin A{**}  & \cellcolor{SeaGreen} \begin{dynkinDiagram}[name=upper]A2
\node (current) at ($(upper root 1)+(0,-.35cm)$) {};
\dynkin[at=(current),name=lower]A2
\begin{pgfonlayer}{Dynkin behind}
\foreach \i in {1,2}{%
\draw[/Dynkin diagram/fold style] ($(upper root \i)$) -- ($(lower root \i)$);}
\end{pgfonlayer}
\end{dynkinDiagram}& \cellcolor{SeaGreen} \dynkin A{*ooo*} & \cellcolor{SeaGreen} \dynkin E{*oooo*}\\ \hline
\cellcolor{lightgray!90} $\mathbb{H}$  &  \cellcolor{RoyalBlue!50} \dynkin C{***}  & \cellcolor{RoyalBlue!50} \dynkin[fold] A{*****} & \cellcolor{RoyalBlue!50} \dynkin D{*oo*o*} & \cellcolor{RoyalBlue!50} \dynkin E{*oooo**}\\ \hline
\cellcolor{lightgray!90} $\mathbb{O}$ &  \cellcolor{OrangeRed!70} \dynkin F{****}  &  \cellcolor{OrangeRed!70} \begin{dynkinDiagram}[mark=o]E{II} \dynkinRootMark{*}1
\dynkinRootMark{*}3 \dynkinRootMark{*}5 \dynkinRootMark{*}6 \dynkinRootMark{*}2 \dynkinRootMark{*}4 \end{dynkinDiagram}

& \cellcolor{OrangeRed!70} \dynkin[backwards] E{*o**oo*o} & \cellcolor{Red} ${\color{white} \dynkin E{*oooo***}}$\\ \hline
\end{tabular}
\end{center}

\end{document}

Also here are the diagrams as made in Keynote, they are drawn to emphasize the residual property, namely taking a point residue (covering a dot if you are unfamiliar with buildings language) in a picture yields the one above when going from row 4 to 3, row 3 to 2 (row 2 to 1 is different and uses folding)

Answer 1

Connecting diagrams is described in §28 of the Dynkin Diagrams Package documentation.

You can adjust the vertical space between upper and lower by changing the -.3cm in the code below:

\documentclass[12pt]{amsbook}

\usepackage{dynkin-diagrams}

\begin{document}

Is this:
\begin{dynkinDiagram}[name=upper]A2
\node (current) at ($(upper root 1)+(0,-.3cm)$) {};
\dynkin[at=(current),name=lower]A2
\begin{pgfonlayer}{Dynkin behind}
\foreach \i in {1,2}{%
\draw[/Dynkin diagram/fold style] ($(upper root \i)$) -- ($(lower root \i)$);}
\end{pgfonlayer}
\end{dynkinDiagram}
what you want?

\end{document}

Answer 2

This is not an answer, but it won't fit as a comment. Here are some very slight changes to your code which I hope make clearer how it works.

\documentclass[12pt]{amsbook}
\usepackage[dvipsnames,table]{xcolor}
\usepackage{dynkin-diagrams}
\begin{document}
\arrayrulecolor{white}
\makeatletter
\def\rulecolor#1#{\CT@arc{#1}}
\def\CT@arc#1#2{%
  \ifdim\baselineskip=\z@\noalign\fi
  {\gdef\CT@arc@{\color#1{#2}}}}
\let\CT@arc@\relax
\rulecolor{white}
\makeatother
\begin{center}
\newcommand\clrK{\rowcolor{BurntOrange!80}}
\newcommand\clrL{\rowcolor{SeaGreen}}
\newcommand\clrH{\rowcolor{RoyalBlue!50}}
\newcommand\clrO{\rowcolor{OrangeRed!70}}
\newcommand\clrOO{\cellcolor{Red}}
\newcommand\rw[1]{\csname clr#1\endcsname\hd{#1}}
\NewDocumentCommand\hd{om}{
\cellcolor{gray!30}$\IfNoValueF{#1}{\mathbb{#1}\setminus}\mathbb{#2}$}
\tikzset{/Dynkin diagram/fold style/.style={blue!22,ultra thick}}
\let\d=\dynkin
\begin{tabular}{|c|c|c|c|c|}\hline
\hd[A]{B}&\hd{K}&\hd{L}&\hd{H}&\hd{O}\\ \hline
\rw{K}& \d A1  & \d A{*o} & \d C{o*o} & \d F{*ooo} \\ \hline
\rw{L}& \d A{**} & 
\begin{dynkinDiagram}[name=upper]A2
\node (current) at ($(upper root 1)+(0,-.35cm)$) {};
\d[at=(current),name=lower]A2
\begin{pgfonlayer}{Dynkin behind}
\foreach \i in {1,2}{%
\draw[/Dynkin diagram/fold style] ($(upper root \i)$) -- ($(lower root \i)$);}
\end{pgfonlayer}
\end{dynkinDiagram}& 
\d A{*ooo*} & 
\d E{*oooo*} \\ \hline
\rw{H} &
\d C{***} &
\d[fold] A{*****} &
\d D{*oo*o*} &
\d E{*oooo**}\\ \hline
\rw{O} & 
\d F{****} &
\d[o/.style = {solid,draw=black,fill=black}] E{II} & 
\d[backwards] E{*o**oo*o} & 
\clrOO \d E{*oooo***}\\ \hline
\end{tabular}
\end{center}
\end{document}